We review some results on the recent development of statistical inference for high-dimensional linear models. We introduce three debiased LASSO estimators, which are asymptotically normal and thus we can construct statistical inference for low dimensional parameters in high-dimensional setting. In addition, we give a brief introduction to the bootstrap assistant procedures to conduct simultaneous inference based on the debiased LASSO.

In this paper, based on the basic theory and application background of stochastic differential equation and backward stochastic differential equation, and combined with stochastic optimal control theory and option price theory in financial market, we will derive the general form of fully coupled forward backward stochastic differential equations (FBSDEs in short). From the point of view of the solvability of this kind of equations, the existing methodology in the literature are analyzed and discussed, a ``unified framework'' approach is introduced to guarantee the existence and uniqueness of solutions for non-Markovian FBSDEs, and several further properties of FBSDEs are obtained. A linear transformation method in virtue of the non-degeneracy of transformation matrix is introduced for cases that the linear FBSDEs, as an important supplement and improvement of the ``unified approach'' method, which makes the application of FBSDEs more extensive.

We present a weighted version of the second Borel-Cantelli lemma, which complements two earlier generalizations of this lemma.

The reliability of k/n system with two interrelated competitive failure mechanisms leading to component failure is considered. The competitive correlation is connected by Gumbel-Barnett (GB) Copula with Weibull marginal distributions that have different scale
parameters and shape parameters. The effects of GB Copula on the failure rate and reliability of components and k/n system are studied when the shape parameters are equal. We proved that the components have increasing, decreasing and bathtub failure rate in terms of the same shape parameter but different values. The reliability of the k/n (n>1) system is not consistently superior to that of the simple system (i.e., k/n=1/1).Based on progressively Type-II censored samples, the Bayes estimation of the parameters and reliability of the k/n system are discussed, and Monte Carlo experiments are carried out to illustrate proposed methods. Finally, a real-life data is provided to demonstrate the model and method proposed in this paper.

In this paper, the Lotka-Volterra stochastic predator-prey systems in a polluted environment is considered. The existence and global attractivity of the periodic solutions are obtained. Furthermore, we obtain the sufficient conditions for the existence and global attractivity of a nontrivial positive periodic solution. Finally, simulations verify our analytical results.

In this paper, we consider the change-point problems for the functional linear model, where the explanatory variable is a random process, and the response is a scalar. Based on the projecting moment estimators of the parameters onto the truncated finite-dimensional space, we propose the detecting statistic and give the estimator of the change-point. In a theoretical investigation, we derive the asymptotic distribution for the proposed detecting statistic, and establish the consistency of the change-point estimate under some mild conditions. Some simulation studies and a real data analysis are conducted to illustrate the finite performance of the proposed testing methods.

Cox model is one of the most popular semi-parametric regression models in epidemiology, biomedical science and clinical trials. In the modeling process, the observed covariates are usually contaminated, and the pollution factor can be measured, but the pollution function is unknown. Therefore, the direct use of the contaminated covariate for parameter estimation may lead to incorrect statistical inference. Researchers often find the best time for disease treatment, and omission of such survival information may lead to a decrease in the estimated effectiveness. In this paper, an improved estimation of the Cox model with contaminated covariates and auxiliary survival information is studied. Kernel smoothing method is used to calibrate the contaminated covariates, and auxiliary survival information is extracted by grouping for parameter estimation. Then, generalized moment estimation method is used to solve the problem of hyperdimensional equations. The results of simulation analysis and empirical study show that the generalized moment estimation method based on the Cox model with covariate calibration is better than the partial likelihood estimation method and the generalized moment estimation method based on the Cox model with the unadjusted covariates.

This paper studies the time-consistent investment strategy of DC pensions with premium return clauses under partial information. We assume that pension managers only have partial information about stock, that is, they can only observe the price of the stock, but not the rate of return of the stock. The pension has a premium return clause. If the insured person dies during the fund accumulation period, the premium paid by him needs to be refunded. In addition, this article also considers inflation and random wages. First, using the Kalman filter theory, the optimal portfolio problem under the partial information situation is transformed into a problem under the complete information situation. Then, by solving an extended HJB equation, the time-consistent investment strategy and the optimal value function are obtained, and a parametric formula of the mean-variance effective frontier is derived. Finally, using the Monte Carlo method to perform numerical simulations, we analyze the effects of partial information and premium return clauses on the stock investment ratio and effective frontier, and give corresponding economic explanations.

Retrospective premium is a premium rating plan that relies on the actual loss of the insurer during the policy period. It underwrites losses that have occurred in the past. The retrospective premium has been used in the reinsurance model in this paper. When the
optimal criterion is to minimize the risk-adjusted value and the risk capital is measured by TVaR, the optimal ceded function under this model is in the form of stop-loss reinsurance. The minimum risk-adjusted value and the optimal retention have been obtained. Finally, through numerical simulation, we assumed that the loss has the exponential distribution, Pareto distribution and Gamma distribution, and explored the influence of the tax multiplier T and the parameter \rho on the optimal retention and the minimum risk adjusted-value. The results show that when other parameters are constant, as T increases, the optimal retention increases and the minimum risk-adjusted value decreases, and when other parameters are constant, the optimal retention and the minimum risk-adjusted value both increase with the increase of \rho. 

The volatility of asset prices is an important research topic that scholars pay close attention to. In recent years, scholars have proposed many estimation methods for volatility, and studied the consistency and asymptotic normality of the estimators. In this paper, we focuses on the NW type kernel estimator of spot volatility proposed by Kristensen\ucite{26}, points out that there are some errors in the relevant proof process, and derives further the strong consistency and uniform strong consistency of the estimator under some reasonable conditions. 

In this paper, for a widely applicable semi-parametric model, partially linear additive model, we study the estimation of its coefficients and nonparametric functions when responses are censored. For this, a composite quantile regression estimation method based on data augmentation is proposed. This method utilizes the relationship between quantile regression and distribution function to construct the imputation dataset, and the final estimators are obtained by composite quantile regression through iterations. The proposed method relaxes the assumptions of the model, not only has low requirements for initial values of iterations but also allows the case when different types of censoring are present in the same dataset. Numerical simulations
show that the proposed method can accurately estimate the coefficients and nonparametric functions of the censored partially linear additive model. In real data analysis, this paper studies the air quality in Beijing, and measures the effects of PM10 concentration, CO concentration, temperature, air pressure, and dew point on PM2.5 concentration. The results show that the composite quantile regression of the partially linear additive model can describe well the influence of these factors on PM2.5 from the perspective of linear and nonlinear relationships, and the proposed method performs well in the processing of censored data.

This paper considers a risk probability minimization problem for nonstationary discrete-time Markov decision processes, in which the transition probabilities and the reward functions depend on time. Different from the expected reward/cost criteria in the existing literature, the optimality performance here is to minimize the probability that the total rewards do not reach a given profit goal until the first passage time to some target set. Under mild reasonable conditions, we establish the corresponding optimality equations, verify that the sequence of the optimal risk functions is the unique solution to the optimality equations, and prove the existence of an optimal Markov policy.

Consider a nearest-neighbor random walk with certain asymptotically zero drift on the positive half line. Let $M$ be the maximum of an excursion starting from 1 and ending at 0. We study the distribution of M and characterize its asymptotics, which is quite different from those of simple random walks.

For high-dimension spatial regression problems, we propose a effective sparse Bayesian model. By introducing a hierarchical Gaussian Markov random field prior, the model can obtain sparse spatial varying parameters, and meanwhile, it can obtain homogeneous parameters estimation for adjacent spatial regions. We use a fast-converging variational EM algorithm for posterior inference, rather than the traditional sampling-based methods. In the M-step of the algorithm, the optimization can be transformed into a classic adaptive lasso problem by simple deformation. The simulation result demonstrate the better performance of our model both in parameters estimation and variable selection. Finally, the proposed model is used to analyze the impact of the socio-demographic factors on the death rate of the COVID-19 in countries of Europe.

We consider an optimal robust investment problem for a defined contribution DC pension plan with stochastic income and
model uncertainty. In the model, the pension account is allowed to invest into a risky asset and a risk-free asset, and the dynamic of the price of risky asset follows a Heston model. The objective of the problem is to maximize the expected utility of the terminal relative wealth by choosing admissible investment strategies. By using the stochastic control dynamic programming approach, we find the robust optimal investment strategy and the corresponding value function when the utility function has the power or the exponential form, respectively. At last, we show a numerical example to further analyze the theoretical results through the MATLAB software.

Let M be a discrete-time normal martingale satisfying some mild conditions, \mathscr{S}(M)\subset L^2(M)\subset \mathscr{S}^*(M) be the Gel'fand triple constructed from the functionals of M. \mathscr{L} denote the space of continuous linear operators from the testing functional space $\mathscr{S}(M)$ to the generalized functional space \mathscr{S}^*(M). As is known, the usual product in \mathscr{L} may not make sense. However, by using the 2D-Fock transform, one can introduce convolution in \mathscr{L}, then one can try to introduce a Bochner-style integral for \mathscr{L}-valued functions with respect to \mathscr{L}-valued measures in the sense of convolution. This paper just studies such a type of integral. First, a class of \mathscr{L}-valued measures are introduced and their basic properties are examined. Then, an integral of an \mathscr{L}-valued function with respect to an $\mathscr{L}$-valued measure is defined and a dominated convergence theorem is established for this integral. Finally, a convolution measure of two \mathscr{L}-valued measures is also discussed and a Fubini type theorem is proved for this integral.

In this paper, we consider the nonparametric estimation of the ruin probability in a compound Poisson risk model perturbed by diffusion. We approximate the ruin probability based on the complex Fourier series expansion (CFS) method, and use a random sample on claim number and individual claim sizes to construct a nonparametric estimator of the ruin probability. We also perform an error analysis of the estimator under a large sample size, and provide simulation results to verify the effectiveness of this estimation method under a finite sample size.

Longitudinal data is an important type of data that is widely used in sociology, economics, biomedicine, epidemiology and other fields. However, in practical problems, people often encounter the situation that the variable dimension is very high and the variable concerned cannot be directly observed, that is, there is a measurement error. In order to solve such problems, this paper studies the estimation of the longitudinal data order index model with measurement error. Based on local linear method and simulation extrapolation (SIMEX) method, this paper constructs a new method for estimating single-index parameters and nonparametric link functions. The effectiveness of the proposed estimation method is verified by Monte Carlo numerical simulation. Compared with the Naive estimation which ignores the measurement error and the estimation which ignores the intra-individual correlation, the estimation constructed in this paper has a smaller mean square error. Finally, we apply the method in this paper to the actual data analysis of the investment demand of listed companies, and the results show that the measurement error has a significant impact on the parameter estimation in practical problems.

We study the numerical solutions for a class of coupled mean-field forward-backward stochastic differential equations. Under suitable regularity assumptions, a posteriori estimate of forward-backward stochastic differential equation is provided. This posterior estimate indicates that the error of the solution for the forward-backward equation can be controlled by the error of the terminal term. Furthermore, we propose a numerical algorithm based on deep neural network and conduct convergence analysis on the discretization scheme.

Currently, spectroscopy technology is widely used in traditional Chinese medicine analysis. In this paper, from the functional
data analysis perspective, we study outlier detection methods for spectral data, detect outliers, and propose the ``Oja depth detection method''. Simulation studies demonstrate the advantages of the Oja depth detection method. We compare the Oja depth detection method with three existing methods on a Chinese medicine spectral data of 73-dose six-mixture liquid. The results show the proposed Oja depth method is able to detect all six unqualified samples and has the highest accuracy.

In financial market, VaR and ES are applied to measure the risk of asset, portfolio management and margin calculation, which are the international unified standards for bank capital and risk supervision. However, VaR has some certain limitations, ES, as an important risk measurement method, has attracted the attention of financial institutions in recent years. Based on the sublinear expectation theory and G-VaR, this study proposes a new calculation method for ES, and denoted as G-ES. This calculation method can be naturally combined with the back testing of G-VaR. Based on the data of S\&P 500 index and CSI300 index, comparing with other commonly used models, such as historical simulation, AR-GARCH model and POT model based on extreme value theory, it is found that this G-ES method has a good performance within different historical data windows.

The pricing problem of correlation options with exchange rate risk under the regime-switching jump-diffusion model is studied. Under the risk neutral measure, it is assumed that the exchange rate follows the regime-switching mean reversion model and the asset prices follow the regime-switching jump-diffusion models. The pricing formula of the correlation options with exchange rate risk is derived by using the measure transform and Fourier transform method. Moreover, the numerical results of option value are provided by the fast Fourier transform algorithm, and the effects of different models and some important parameters on the value of correlation options with exchange rate risk are analyzed.

The probabilistic combination method is a very useful method to analyze the reliability of wireless sensor network (WSN), and it can effectively deal with the isolation effect and competition failure in the network. However, there are some shortcomings in the reliability modeling and calculation of WSN based on the probabilistic combination method in the existing literature, which leads to a limited application of the research results and even an incorrect evaluation for the reliability of WSN. This paper studies the reliability modeling and calculation of a typical WSN which has only one relay node and the probabilistic functional dependence mechanism. Under the general assumption that WSN faces local failures of components and global failures caused by external attacks, a reliability model that is more realistic and more rigorous is established for WSN. Based on rigorous probability combination analysis, it shows systematic method and compact formulas for the reliability of WSN. The current study enhances the research on such issues in the literature efficiently. Finally, as applications, we calculate the reliability of two special WSNs ---body sensor network and air monitoring system.

This paper concerns the exponential utility maximization problem for semi-Markov decision process with Borel state and action spaces, and nonnegative rewards. The optimal criterion is maximize the expectation of exponential utility of the total rewards
in infinite horizon. Under the regular and compactness-continuity conditions, we establish the corresponding optimality equation, and prove the existence of an exponential utility optimal stationary policy by an invariant embedding technique. Moreover, we provide an iterative algorithm for calculating the value function as well as the optimal policies. Finally, we illustrate the computational aspects of an optimal policy with an example. 

In this paper we study a discrete-time Geo/T-IPH/1 queue model, where T-IPH denotes the discrete-time phase type distribution defined on a birth and death process with countably many states. The queue model can be described by a quasi-birth-and-death (QBD) process with countably many phases. Using operator-geometric solution method, we first give the expression of the joint stationary distribution. Then we obtain the explicit stationary queue length distribution of the queue we considered. Finally, a numerical example is also presented to illustrate the computational procedure.

This study considers the reliability of a multicomponent stress-strength model involving one stress and multiple strengths from a series system. We derive the Jeffreys prior when the stress and strength variables follow Weibull distribution with a common shape parameter. The necessary and sufficient conditions of the propriety of the posterior distribution based on the Jeffreys prior are obtained. Lindley's approximation and Markov chain Monte Carlo method are presented to obtain the estimates of the system reliability. The performance of the proposed methods is evaluated by Monte Carlo simulation. The simulation results show the Bayesian method outperforms maximum likelihood method, especially in the case of a small sample size. Finally, a real dataset is analyzed for illustration.

This paper mainly proposes Bayesian methods to analyze the accelerated failure time models. In this model, the distribution of the error terms is unknown and approximated with a P\'{o}lya tree distribution. This paper employs the Bayesian Lasso and Markov chain Monte Carlo methods for parameter estimation and variable selection. Simulation studies demonstrate that the proposed methods can identify the important factors and provide accurate estimates. Finally, the proposed model is applied to identify the risk factors of survival times of the Type II diabetic patients.

Semi-supervised data contains a labeled data set with both responses and covariates and an unlabeled data set with covariates only. The inference based on semi-supervised data is gaining more and more interests in statistics. When the response in the labeled data is binary, case-control sampling is commonly used to alleviate the imbalanced data structure. When the response and the covariates satisfy the logistic model, the slope parameter of the model can be consistently estimated even for the case-control sampling. However, when the logistic model is incorrectly specified for the data, the case-control samples can not estimate the population risk minimizer consistently. With the help of the unlabeled data, we derive a consistent estimator for the case population proportion. Then, an inverse probability weighted loss function is developed to obtain a consistent estimator for the population risk minimizer. The proposed estimators are shown to be asymptotically normal and the limiting variance-covariance matrix can be consistently estimated. Simulation results show that the proposed method gives out reasonable finite sample performances. A real data example is also analyzed for illustration.

With the advance of the era of big data, high-dimensional data are frequently collected in many research fields such as economics, finance, and biomedicine. One of the characteristics of high-dimensional data is that the variable dimension p increases with the increase of the sample size n and usually exceeds the sample size. At the same time, outliers are also prone to appear in high-dimensional data. Therefore, how to overcome the influence of outliers on high-dimensional statistical inference, so as to obtain a
more accurate model, is one of the hot issues in current statistical research. This paper is an overview of robust variable selection methods under high-dimensional linear models. Specifically, first of all, we introduce three indicators to evaluate robustness: influence function, breakdown point and maximum deviation. Secondly, it focuses on the selection methods of robust variables, including response variables with outliers, response variables and covariates with outliers, high breakdown point and efficient variable selection methods. Then, the related algorithms are introduced, and different variable selection methods are compared through simulation and examples. Finally, the problems of high-dimensional robust effective variable selection methods and the possible development direction in the future are briefly discussed.

Let \{X,X_n,n\geq1\} be a sequence of identically distributed NA random variables and set S_n=\sum_{i=1}^nX_i, n\geq 1. Let h(\cdot) be a positive nondecreasing function on (0,\infty) such that \int_1^\infty[th(t)]^{-1}\md t=\infty. Denote Lt=\ln\max\{e,t\}, S_n=\sum_{i=1}^nX_i, \psi(t)=\int_1^t[sh(s)]^{-1}\md s, t\geq 1. In this paper, we prove that \sum_{n=1}^\infty[nh(n)]^{-1}\pr(\max_{1\leq j\leq n}|S_j|\geq (1+\varepsilon)\sigma\sqrt{2nL\psi(n)})<\infty, \forall\,\varepsilon>0 if and if \ep(X)=0 and \ep(X^2)=\sigma^2\in(0,\infty). The result partially extends and improves the theorems of \ncite{7}.

Precision medicine emphasizes the importance of correctly identifying heterogeneous subgroups to develop individualized treatment strategies that can be prescribed for each subgroup. Despite the recent methodological advances in subgroup analysis, how to effectively identify subgroups for censored data remains largely unexplored. In this paper, we propose a new methodology for subgroup analysis based on the accelerated failure time model, in which the heterogeneity caused by latent factors can be represented by subject-specific intercepts. We consider the most common right censoring situation, and processes the censored data by the mean imputation method. The hard thresholding penalty function is applied to pairwise differences of the intercepts, thus automatically dividing the observations into different subgroups. We also establish the theoretical properties of our proposed estimator. Our proposed method is further illustrated by simulation studies and analysis of a wisconsin breast cancer dataset.

We consider quadratic weighted branching processes with immigration and instantaneous resurrection. Under the condition that
the resurrection rate can be summed, we show that the target process does not exist. The existence and uniqueness criterion for the process are obtained under the assumption that the sum of the resurrection is infinite. Also, the equivalent conditions for the existence criterion are given for easy verification. It is proved that there exist infinitely many of Q processes. Among them, there exists a unique honest process and the corresponding construction method is then investigated. We prove that this honest process is always ergodic and the second-order differential equation of the equilibrium distribution is established.

Simple random sampling, both with and without replacement, are fundamental in traditional survey sampling. Based on an idea of ``imaginary census'', we provide in this note a new way for calculating the sample variance of simple sample average, as well as understanding the intrinsic relationship between simple random sampling with and without replacement. The key concept is an ``imaginary census'' matrix, which records the exact sampling trajectory of each draw without replacement, until all population units were sampled out. The random matrix possesses a nice probabilistic symmetry, and each of its column summed to be a fixed number. The new framework, in a sense, is a fusion of two existing classic techniques in traditional survey sampling. One depends on the random vector of 0\,--\,1 valued random variables indicating which population units were sampled, and the other is the symmetrization technique. Our method appears valuable for understanding several important sampling strategies, even to the branch of survey sampling itself. For illustration, we present two examples pertain to unequal probability sampling with replacement and adaptive cluster sampling, with a focus on understanding these sampling strategies from the perspective of simple random sampling.

In this paper, we study a class of one-dimensional time-inhomogeneous stochastic differential equations with mean field. We
show that the unique solution is ergodic under certain conditions. We further show that, as the strength of the mean field tends to 0, the solution and stationary distribution of such equation respectively converge a.e. \!\!uniformly and in Wasserstein distance to those of corresponding equation without mean field.

Component orthogonal arrays is a kind of experimental designs that are suitable for order-of-addition experiments. This paper utilizes mutually orthogonal Latin squares obeying certain restrictions to construct component orthogonal arrays where the number of factors is a power of a prime. Further more, construction methods for component orthogonal arrays with flexible number of factors are provided. Some favorable projection properties of the obtained designs are demonstrated.

The actuarial calculation of annuities is closely related to the interest rate model. In standard annuities, the interest rate for each period is a fixed constant. In practice, the interest rate for each period can be a variable or even a random variable. These random
variables constitute a stochastic process of interest rate. In many cases, the stochastic process of interest rate is a Markov process. This article studies the actuarial calculation of annuities under the Markov stochastic interest rate model. It is proved that if the interest rate process is a time-homogeneous Markov chain, then the discounting process is also a time-homogeneous Markov chain, and they have the `same' initial distribution and the `same' one-step transition probability matrix. With the help of the interest rate discounting process, the expectation and variance of the present value of annuities under the Markov stochastic interest rate model are calculated. This article introduced annuity polynomials, operators, and annuity operator polynomials. It makes the expressions of expectation and variance for annuities very concise, and easy to program and calculate.